Make the upgrade 2.0 -> 2.1.20 then you'll be able to upgrade to 3.11.3 i.e. 2.1.20 -> 3.11.3. You can upgrade to 3.0.17 as an intermediary step (I would do), but don't upgrade to 2.2. Also make sure to read carefully https://github.com/apache/cassandra/blob/cassandra-3.11/NEWS.txt It's a long read but it's important. There are lots of changes between all these versions.

We have a pretty outdated Cassandra cluster running version 2.0.x. Instead of doing step by step upgrades (2.0 -> 2.1, 2.1 -> 2.2, 2.2 -> 3.0, 3.0 -> 3.11.x), would it be possible to add new nodes with a recent version (say 3.11.x) and start decommissioning the old ones until we have a cluster with only 3.11.x?